Silting subcategories and (co)torsion pairs
associated to extended hearts
June 11, 2026
We establish the poset isomorphisms between \((d+1)\)-term silting subcategories, functorially finite \(s\)-torsion pairs in the \(d\)-extended heart, and hereditary complete cotorsion pairs in a suitable subcategory. As an application, we also give dg algebra versions of these bijections, which establish the poset isomorphisms between \(\tau\)-tilting pairs, \((d+1)\)-term silting complexes, and functorially finite \(s\)-torsion pairs.
-0.5Introduction Support \(\tau\)-tilting modules were introduced by Adachi, Iyama and Reiten [1] to complete the classical tilting theory via mutations. Furthermore, they established bijections between support \(\tau\)-tilting pairs, functorially finite torsion pairs, and \(2\)-term silting complexes. Pauksztello and Zvonareva [2] gave a bijection between functorially finite torsion pairs and complete cotorsion pairs in a certain \(2\)-term category. In order to generalize these bijections to silting complexes of arbitrary length, the \(d\)-extended module categories were introduced in [3], [4]. More general, the \(d\)-extended heart was introduced in [4], which is a natural generalization of the hearts of \(t\)-structures in triangulated categories and forms an extriangulated category in the sense of [5]. The \(s\)-torsion pairs in extriangulated categories were introduced in [6] with the aim to provide a general framework for the study of \(t\)-structures in triangulated categories and torsion pairs in abelian categories. Recently, Gupta [3] generalized the bijections above to a more general framework, and established poset isomorphisms between \((d+1)\)-term silting subcategories, functorially finite \(s\)-torsion pairs in \(d\)-extended hearts, and hereditary complete cotorsion pairs in appropriate subcategories. Zhou [4] proved that \(d\)-extended hearts admit almost split extensions, introduced \(\tau\)-tilting pairs in \(d\)-extended hearts, and generalized the bijections above to \(d\)-extended hearts, thereby expanding the application scope of tilting theory.
In addition to the case of finite-dimensional algebras, Wei and Zhou [7] extended their research perspective to more general triangulated categories. They introduced the concept of AIR tilting subcategories as generalizations of support \(\tau\)-tilting modules, and established a bijection between AIR tilting subcategories and \((d+1)\)-term silting subcategories under appropriate conditions.
In this paper, we generalize Gupta’s bijections by establishing poset isomorphisms between \((d+1)\)-term silting subcategories, functorially finite \(s\)-torsion pairs in \(d\)-extended hearts, and hereditary complete cotorsion pairs in specific subcategories under a more general Krull-Schmidt triangulated category framework. In the framework of non-positive dg algebras, Mochizuki and Plogmann [13] proved that each \(d\)-extended heart admits ARS duality in the sense of [10], provided that the dg algebra is contained in its \(d\)-extended heart as a regular dg module. This enables us to extend the \(\tau\)-tilting theory to derived categories of non-positive dg algebras and obtain a dg analogue of the bijections above.
Throughout the paper, all categories are additive categories. Let \(\mathcal{C}\) be a category, for any collection \(\mathcal{X}\) consisting of some objects in \(\mathcal{C}\), we denote by \(\operatorname{add}\mathcal{X}\) the additive closure of \(\mathcal{X}\) in \(\mathcal{C}\), which is defined to be the full subcategory of \(\mathcal{C}\) consisting of direct summands of finite direct sums of objects in \(\mathcal{X}\). We write \(\operatorname{add}(X)\) as \(\operatorname{add}\mathcal{X}\) if \(\mathcal{X}\) consists of a single object \(X\). When we say that \(\mathcal{X}\) is a subcategory of \(\mathcal{C}\), we always assume that \(\mathcal{X}\) is full and satisfies \(\mathcal{X} =\operatorname{add}\mathcal{X}\). For any two subcategories \(\mathcal{X}\) and \(\mathcal{Y}\) of \(\mathcal{C}\), we write \(\operatorname{Hom}_{\mathcal{C}}(\mathcal{X},\mathcal{Y}) = 0\) to mean that \(\operatorname{Hom}_{\mathcal{C}}(X,Y) = 0\) for any \(X \in \mathcal{X}\) and \(Y \in \mathcal{Y}\). The analogous conventions apply to the notations \(\operatorname{Hom}_{\mathcal{C}}(\mathcal{X},Y) = 0\) and \(\operatorname{Hom}_{\mathcal{C}}(X,\mathcal{Y}) = 0\) for any fixed objects \(X\) and \(Y\) in \(\mathcal{C}\). For the sake of simplicity in notation, we may write \(\operatorname{Hom}(X,Y)\) as \(\operatorname{Hom}_{\mathcal{C}}(X,Y)\) for any \(X,Y \in \mathcal{C}\).
-0.5Preliminaries In this section, we set some notations and recall some fundamental notions. Let \(\mathcal{C}\) be a triangulated category with the suspension functor denoted by \([1]\). Let \(I\) be a subset of the integer set \(\mathbb{Z}\), which is often taken to be the subset denoted by \(>n\), \(<n\), \(\geq n\), \(\leq n\), \(\neq n\) or \(n\) with the obvious associated meaning. For any subcategory \(\mathcal{X}\) of \(\mathcal{C}\), we define the following classes \[\begin{align} &\mathcal{X}[I]:=\{X[i]\,|\,X\in\mathcal{X}\text{ and }i\in I\},\\ &{^{\perp_I}\mathcal{X}} := \{Y \in \mathcal{C}\,|\,\operatorname{Hom}(Y,\mathcal{X}[I]) = 0\},\\ &\mathcal{X}^{\perp_I} := \{Y \in \mathcal{C}\,|\,\operatorname{Hom}(\mathcal{X},Y[I]) = 0\}. \end{align}\] If \(\mathcal{X}={\rm add}(X)\) for some object \(X\in\mathcal{C}\), we write \({^{\perp_I}X}\) and \(X^{\perp_I}\) for \({^{\perp_I}\mathcal{X}}\) and \(\mathcal{X}^{\perp_I}\), respectively. For any subcategories \(\mathcal{X}\) and \(\mathcal{Y}\) of \(\mathcal{C}\), we denote by \(\mathcal{X} \ast \mathcal{Y}\) the class of all objects \(Z\) admitting a triangle \(X \to Z \to Y \to X[1]\) with \(X \in \mathcal{X}\) and \(Y \in \mathcal{Y}\).
In what follows, let \(\mathcal{B}\) be an extension-closed subcategory of \(\mathcal{C}\), which means that for any triangle \(X \to Z \to Y \to X[1]\) with \(X,Y\in\mathcal{B}\), we have \(Z\in\mathcal{B}\). Then \(\mathcal{B}\) is an extriangulated category in the sense of [5].
Definition 1. A pair of subcategories \((\mathcal{T},\mathcal{F})\) of \(\mathcal{B}\) is said to be a torsion pair if \({\rm Hom}(\mathcal{T},\mathcal{F})=0~\text{and}~\mathcal{B}=\mathcal{T}\ast\mathcal{F}.\) In this case, the subcategory \(\mathcal{T}\) is called the torsion class, and \(\mathcal{F}\) is called the torsion-free class. The torsion pair \((\mathcal{T},\mathcal{F})\) is called an \(s\)-torsion pair if moreover \({\rm Hom}(\mathcal{T},\mathcal{F}[\leq -1])=0\). We denote by \(\operatorname{tors}\mathcal{B}\) the poset of torsion classes in \(\mathcal{B}\) under inclusion, and denote by \(s\operatorname{-tors}\mathcal{B}\) the poset of \(s\)-torsion classes in \(\mathcal{B}\) under inclusion.
****Remark** 1**. The torsion pairs give rise to approximations. Explicitly, let \((\mathcal{T},\mathcal{F})\) be a torsion pair in \(\mathcal{B}\). Then every \(X\in\mathcal{B}\) admits a triangle \[{T\stackrel{f}{\longrightarrow}X\stackrel{g}{\longrightarrow}F\stackrel{}{\longrightarrow}T[1]}\] with \(T\in\mathcal{T}\) and \(F\in\mathcal{F}\), which implies \(f\) is a right \(\mathcal{T}\)-approximation of \(X\) and \(g\) is a left \(\mathcal{F}\)-approximation of \(X\). In particular, \(\mathcal{T}\) is contravariantly finite and \(\mathcal{F}\) is covariantly finite in \(\mathcal{B}\). Moreover, \(\mathcal{T}={^{\bot_0}\mathcal{F}}\cap\mathcal{B}\) and \(\mathcal{F}=\mathcal{T}^{\bot_0}\cap\mathcal{B}\).
Definition 2. A pair of subcategories \((\mathcal{U},\mathcal{V})\) is called a cotorsion pair in \(\mathcal{B}\) if
\({\rm Hom}(\mathcal{U},\mathcal{V}[1])=0;\)
For any \(B\in\mathcal{B}\), there exist two triangles \[V\stackrel{}{\longrightarrow} U\stackrel{}{\longrightarrow} B\stackrel{}{\longrightarrow} V[1] \text{ and } B\stackrel{}{\longrightarrow} V'\stackrel{}{\longrightarrow} U'\stackrel{}{\longrightarrow} B[1]\] such that \(U,\,U'\in\mathcal{U}\) and \(V,\,V'\in\mathcal{V}\).
In this case, the subcategory \(\mathcal{V}\) is called the cotorsion class. We denote by \(\operatorname{cotors}\mathcal{B}\) the poset of cotorsion classes in \(\mathcal{B}\) under inclusion.
The cotorsion pair and the corresponding cotorsion class are said to be hereditary if moreover \({\rm Hom}(\mathcal{U},\mathcal{V}[\geq 2])=0\). We denote by \(\operatorname{h.cotors}\mathcal{B}\) the poset of hereditary cotorsion classes in \(\mathcal{B}\) under inclusion.
****Remark** 2**. Let \((\mathcal{U},\mathcal{V})\) be a cotorsion pair in \(\mathcal{B}\). Similarly to torsion pairs, we have that \(\mathcal{U}\) is contravariantly finite and \(\mathcal{V}\) is covariantly finite in \(\mathcal{B}\). Moreover, \(\mathcal{U}={^{\bot_1}\mathcal{V}}\cap\mathcal{B}\) and \(\mathcal{V}=\mathcal{U}^{\bot_1}\cap\mathcal{B}\).
Definition 3. A torsion pair \((\mathcal{T},\mathcal{F})\) in \(\mathcal{C}\) is said to be
a \(t\)-structure if \(\mathcal{T}[1]\subseteq\mathcal{T}\), in this case, \(\mathcal{T}\cap\mathcal{F}[1]\) is called the heart of \((\mathcal{T},\mathcal{F})\);
a co-\(t\)-structure if \(\mathcal{T}[-1]\subseteq\mathcal{T}\), in this case, \(\mathcal{T}[1]\cap\mathcal{F}\) is called the coheart of \((\mathcal{T},\mathcal{F})\);
nondegenerate if \(\bigcap_{k\in \mathbb{Z}}\mathcal{T}[k]=\{0\}=\bigcap_{k\in \mathbb{Z}}\mathcal{F}[k]\);
bounded if \(\bigcup_{k\in \mathbb{Z}}\mathcal{T}[k]=\mathcal{C}=\bigcup_{k\in \mathbb{Z}}\mathcal{F}[k]\).
****Remark** 3**. \((1)\) By [6], a torsion pair \((\mathcal{T},\mathcal{F})\) in \(\mathcal{C}\) is a \(t\)-structure if and only if it is an \(s\)-torsion pair in \(\mathcal{C}\). We usually denote a \(t\)-structure in \(\mathcal{C}\) by \((\mathcal{C}^{\leq 0},\mathcal{C}^{\geq 1})\), and set \[(\mathcal{C}^{\leq n},\mathcal{C}^{\geq n+1}):=(\mathcal{C}^{\leq 0}[-n],\mathcal{C}^{\geq 1}[-n])\] for any \(n\in\mathbb{Z}\). For any \(X\in\mathcal{C}\), there exists a unique triangle up to isomorphism \[{\tau_{\leq n}X\stackrel{}{\longrightarrow} X\stackrel{}{\longrightarrow}\tau_{\geq n+1}X\stackrel{}{\longrightarrow}\tau_{\leq n}X[1]}\] such that \(\tau_{\leq n}X\in\mathcal{C}^{\leq n}\) and \(\tau_{\geq n+1}X\in\mathcal{C}^{\geq n+1}\), which is called the \(t\)-decomposition of \(X\) with respect to the \(t\)-structure \((\mathcal{C}^{\leq n},\mathcal{C}^{\geq n+1})\). This \(t\)-decomposition induces truncated functors \[\tau_{\leq n}:\mathcal{C}\longrightarrow\mathcal{C}^{\leq n} \text{ and } \tau_{\geq n+1}:\mathcal{C}\longrightarrow\mathcal{C}^{\geq n+1}\] such that \((\sigma_{\leq n},\tau_{\leq n})\) and \((\tau_{\geq n+1},\sigma_{\geq n+1})\) are adjoint pairs, where \(\sigma_{\leq n}:\mathcal{C}^{\leq n}\to\mathcal{C}\) and \(\sigma_{\geq n+1}:\mathcal{C}^{\geq n+1}\to\mathcal{C}\) are the natural inclusion functors.
\((2)\) A torsion pair \((\mathcal{T},\mathcal{F})\) in \(\mathcal{C}\) is a co-\(t\)-structure if and only if \((\mathcal{T}[1],\mathcal{F})\) is a hereditary cotorsion pair in \(\mathcal{C}\).
Definition 4. Let \(\mathcal{S}\) be a subcategory of \(\mathcal{C}\).
\((1)\) The subcategory \(\mathcal{S}\) is called presilting if \({\rm Hom}(\mathcal{S},\mathcal{S}[>0])=0\).
\((2)\) The subcategory \(\mathcal{S}\) is called silting if it is presilting and \({\boldsymbol{t}hick}_{\mathcal{C}}(\mathcal{S})=\mathcal{C}\), where \({\boldsymbol{t}hick}_{\mathcal{C}}(\mathcal{S})\) denotes the smallest triangulated subcategory of \(\mathcal{C}\) which contains \(\mathcal{S}\) and is closed under direct summands.
Let us define a partial order on the collection \(\operatorname{silt}\mathcal{C}\) of silting subcategories of \(\mathcal{C}\) as follows: \[\label{siltingorder} \text{For~any}~\mathcal{S}_1,\mathcal{S}_2\in\operatorname{silt}\mathcal{C},~\text{define}~\mathcal{S}_1\leq \mathcal{S}_2\Longleftrightarrow{\rm Hom}(\mathcal{S}_2,\mathcal{S}_1[>0])=0.\tag{1}\] Let \(\mathcal{S}\) be a silting subcategory of \(\mathcal{C}\). For any integers \(p\leq q\), set \[\begin{align} &\mathcal{S}^{[p,q]}:=\mathcal{S}[p]\ast\mathcal{S}[p+1]\ast\cdots\ast\mathcal{S}[q],\\ &\mathcal{S}^{(-\infty,q]}:=\bigcup_{k = -q}^{+\infty} \mathcal{S}^{[-k,q]},\\ &\mathcal{S}^{[p,+\infty)}:=\bigcup_{k = p}^{+\infty} \mathcal{S}^{[p,k]}. \end{align}\]
Lemma 1. Let \(\mathcal{S}_1,\mathcal{S}_2\) be presilting subcategories of \(\mathcal{C}\) such that \({\boldsymbol{t}hick}_{\mathcal{C}}(\mathcal{S}_1)={\boldsymbol{t}hick}_{\mathcal{C}}(\mathcal{S}_2)\). Then for any integers \(p\leq q\), the following are equivalent:
\(\mathcal{S}_2\subseteq \mathcal{S}_1^{[p,q]};\)
\(\mathcal{S}_1[q]\leq \mathcal{S}_2\leq\mathcal{S}_1[p];\)
\(\mathcal{S}_1^{\perp_{>0}}[q]\subseteq \mathcal{S}_2^{\perp_{>0}}\subseteq \mathcal{S}_1^{\perp_{>0}}[p]\).
Proof. The equivalence of \((1)\) and \((2)\) is obtained by [8]. The implication \((3)\Rightarrow (2)\) follows from \(\mathcal{S}_1\subseteq \mathcal{S}_1^{\perp_{>0}}\) and \(\mathcal{S}_2\subseteq \mathcal{S}_2^{\perp_{>0}}\).
For \((1)\Rightarrow (3)\), since \(\mathcal{S}_1^{\perp_{>0}}[q]=(\mathcal{S}_1[q])^{\perp_{>0}}=(\mathcal{S}_1[q-1])^{\perp_{>-1}}=\cdots=(\mathcal{S}_1[p])^{\perp_{>p-q}},\) we have \(\mathcal{S}_1^{\perp_{>0}}[q]\subseteq(\mathcal{S}_1^{[p,q]})^{\perp_{>0}}\subseteq\mathcal{S}_2^{\perp_{>0}}\). On the other hand, by [8], we get \(\mathcal{S}_1\subseteq \mathcal{S}_2^{[-q,-p]}\). Thus, \(\mathcal{S}_2^{\perp_{>0}}[-p]\subseteq(\mathcal{S}_2^{[-q,-p]})^{\perp_{>0}}\subseteq\mathcal{S}_1^{\perp_{>0}}\). Hence, we finish the proof. ◻
-0.5Bijections between silting subcategories, \(s\)-torsion and cotorsion pairs In this section, let \(\mathcal{D}\) be a triangulated category with a presilting subcategory \(\mathcal{P}\) such that \[t=(\mathcal{D}^{\leq 0}, \mathcal{D}^{\geq 1}):=(\mathcal{P}^{\perp_{>0}}, \mathcal{P}^{\perp_{\leq 0}})\] is a \(t\)-structure in \(\mathcal{D}\) with the heart denoted by \(\mathcal{H}^1_{\mathcal{P}}\). Fix a positive integer \(d\), we define the \(d\)-extended heart to be \[\mathcal{H}_{\mathcal{P}}^{d}:=\mathcal{D}^{\leq 0}\cap\mathcal{D}^{\geq -d+1}.\] Note that both \(\mathcal{D}^{\leq 0}\) and \(\mathcal{D}^{\geq -d+1}\) are closed under extensions, then so is \(\mathcal{H}_{\mathcal{P}}^{d}\). Thus, \(\mathcal{H}_{\mathcal{P}}^{d}\) forms an extriangulated category, whose biadditive functor \(\mathbb{E}\) is defined by \[\mathbb{E}(X,Y) := \operatorname{Hom}(X, Y[1])\] for any \(X,Y \in \mathcal{H}_{\mathcal{P}}^{d}\), and the sequence \[Y\stackrel{f}{\longrightarrow}Z\stackrel{g}{\longrightarrow}X\stackrel{\delta}\dashrightarrow\] with \(X,Y,Z \in \mathcal{H}_{\mathcal{P}}^{d}\) is an extriangle in \(\mathcal{H}_{\mathcal{P}}^{d}\) if \[{Y \stackrel{f}{\longrightarrow} Z \stackrel{g}{\longrightarrow} X \stackrel{\delta}{\longrightarrow} Y[1]}\] is a triangle in \(\mathcal{D}\). Similarly, \(\mathcal{P}^{[0,d]}\) is also an extriangulated category.
Set \(\mathcal{K}(\mathcal{P}):={\boldsymbol{t}hick}_{\mathcal{D}}(\mathcal{P})\). By definition, \(\mathcal{P}\) is a silting subcategory of \(\mathcal{K}(\mathcal{P})\). Let \(\operatorname{b.co-t-str}\mathcal{K}(\mathcal{P})\) be the poset of bounded co-\(t\)-structures in \(\mathcal{K}(\mathcal{P})\) ordered by the inclusions of torsion-free classes.
****Proposition** 4**. [9]There is an isomorphism of posets \[\begin{align} \chi : \rm silt\,\mathcal{K}(\mathcal{P}) \longrightarrow \operatorname{b.co-t-str}\mathcal{K}(\mathcal{P}),~ \mathcal{S} \longmapsto (\mathcal{S}^{(-\infty,-1]},\mathcal{S} ^{[0,+\infty]}). \end{align}\] Moreover, \(\mathcal{S} ^{[0,+\infty]}=\mathcal{S}^{\perp_{>0}}\cap\mathcal{K}(\mathcal{P}).\)
Let \((d+1)\)-\({\rm silt}\,\mathcal{P}\) be the poset of the silting subcategories of \(\mathcal{K}(\mathcal{P})\) contained in \(\mathcal{P}^{[0,d]}\) under the order defined as 1 . In what follows, each subcategory in \((d+1)\)-\({\rm silt}\,\mathcal{P}\) is called the \((d+1)\)-term silting subcategory with respect to \(\mathcal{P}\).
****Proposition** 5**. There is an isomorphism of posets \[\begin{align} \psi : (d+1)\operatorname{-silt}\mathcal{P} \longrightarrow \operatorname{h.cotors}\mathcal{P}^{[0,d]},~ \mathcal{S} \longmapsto \mathcal{S} ^{[0,+\infty]} \cap \mathcal{P}^{[0,d]}. \end{align}\]
Proof. By Proposition 4 and Lemma 1, the map \(\chi\) restricts to a poset isomorphism: \[\chi : (d+1)\operatorname{-silt}\mathcal{P} \stackrel{\sim}{\longrightarrow} \operatorname{b.co-t-stfs}\,[0,d],\] where \(\operatorname{b.co-t-stfs}\,[0,d]\) denotes the poset \[\{\mathcal{Y}~|~({^{\perp_{0}}}\mathcal{Y}\cap\mathcal{K}(\mathcal{P}),\mathcal{Y}) \,\text{is a bounded co-}t\text{-structure in} \,\mathcal{K}(\mathcal{P})~\text{s.t.}~\mathcal{P}^{[d,+\infty)}\subseteq \mathcal{Y}\subseteq \mathcal{P}^{[0,+\infty)}\}\] under inclusion. Note that \(\mathcal{P}^{[0,d]}=\mathcal{P}^{(-\infty,d]}\cap\mathcal{P}^{[0,+\infty)}\). Applying [10] to \((\mathcal{P}^{(-\infty,d]},\mathcal{P}^{[d+1,+\infty)})\) and \((\mathcal{P}^{(-\infty,-1]},\mathcal{P}^{[0,+\infty)})\), we obtain a poset isomorphism \[\begin{align} \xi: \operatorname{cotors}\,[0,d] \stackrel{\sim}{\longrightarrow} \operatorname{cotors}\mathcal{P}^{[0,d]},~ \mathcal{\mathcal{Y}}\longmapsto \mathcal{Y} \cap \mathcal{P}^{[0,d]} \end{align}\] with the inverse map \(\eta\) defined by \(\mathcal{\mathcal{V}} \mapsto \operatorname{add}(\mathcal{V}\ast \mathcal{P}^{[d,+\infty)})\), where \(\operatorname{cotors}\,[0,d]\) denotes the poset \[\{\mathcal{Y}~|~({^{\perp_{1}}}\mathcal{Y}\cap\mathcal{K}(\mathcal{P}),\mathcal{Y})~\text{is a cotorsion pair in} ~\mathcal{K}(\mathcal{P}) \text{ s.t.}~\mathcal{P}^{[d,+\infty)}\subseteq \mathcal{Y}\subseteq \mathcal{P}^{[0,+\infty)}\}\] under inclusion and \({^{\perp_1}}\eta(\mathcal{V} )\cap\mathcal{K}(\mathcal{P})=\operatorname{add}(\mathcal{P}^{(-\infty,0]}\ast ({^{\perp_1}\mathcal{V}}\cap\mathcal{P}^{[0,d]}))\).
For any \((d+1)\)-term silting subcategory \(\mathcal{S}\), the image \(\xi (\chi (\mathcal{S}))=\mathcal{S} ^{[0,+\infty]} \cap \mathcal{P}^{[0,d]}\) is hereditary. Denote by \(\xi'\) the restriction of \(\xi\) on \(\operatorname{b.co-t-stfs}\,[0,d]\) and set \(\psi=\xi'\circ\chi\), i.e. we have the following commutative diagram \[\xymatrix{ (d+1)\operatorname{-silt}\mathcal{P} \ar[rr]^-{\chi}_-{\cong} \ar[dr]_-{\psi} & & \operatorname{b.co-t-stfs}\,[0,d]\ar@{>->}[dl]^-{\xi'}\\ & \operatorname{h.cotors}\mathcal{P}^{[0,d]} }\]
Let \((\mathcal{U}, \mathcal{V})\) be a hereditary cotorsion pair in \(\mathcal{P}^{[0,d]}\). Then \[(\operatorname{add}(\mathcal{P}^{(-\infty,0]}\ast \mathcal{U}), \operatorname{add}(\mathcal{V}\ast \mathcal{P}^{[d,+\infty)}))\] is a cotorsion pair satisfying \[\mathcal{P}^{[d,+\infty)}\subseteq \operatorname{add}(\mathcal{V}\ast \mathcal{P}^{[d,+\infty)})\subseteq \mathcal{P}^{[0,+\infty)}\] and \(\xi(\operatorname{add}(\mathcal{V}\ast\mathcal{P}^{[d,+\infty)}))= \xi(\eta(\mathcal{V}))=\mathcal{V}\).
By the heredity of \((\mathcal{U},\mathcal{V})\), we obtain \[(\operatorname{add}(\mathcal{P}^{(-\infty,0]}\ast \mathcal{U}), \operatorname{add}(\mathcal{V}\ast \mathcal{P}^{[d,+\infty)}))\] is hereditary. Thus, by Remark 3 \((2)\), we conclude that \[\label{ytjg} (\operatorname{add}(\mathcal{P}^{(-\infty,0]}\ast \mathcal{U})[-1], \operatorname{add}(\mathcal{V}\ast \mathcal{P}^{[d,+\infty)}))\tag{2}\] is a co-\(t\)-structure in \(\mathcal{K}(\mathcal{P})\). Since \[\begin{align} \bigcup_{n \in \mathbb{Z}} \mathrm{add}( \mathcal{P}^{(-\infty,0]} \ast \mathcal{U})[n] \supseteq \bigcup_{n \in \mathbb{Z}} \mathcal{P}^{(-\infty,n]} = \mathcal{K} (\mathcal{P}),\\ \bigcup_{n \in \mathbb{Z}} \mathrm{add}( \mathcal{V} \ast \mathcal{P}^{[0,+\infty)})[n] \supseteq \bigcup_{n \in \mathbb{Z}} \mathcal{P}^{[n,+\infty)} = \mathcal{K} (\mathcal{P}), \end{align}\] we get that the co-\(t\)-structure 2 is bounded. Thus, we obtain \[\mathrm{add}( \mathcal{V} \ast \mathcal{P}^{[0,+\infty)})\in \operatorname{b.co-t-stfs}\,[0,d]\] and \[\xi'(\operatorname{add}(\mathcal{V}\ast\mathcal{P}^{[d,+\infty)}))=\xi(\operatorname{add}(\mathcal{V}\ast\mathcal{P}^{[d,+\infty)}))= \mathcal{V}.\] That is, \(\xi'\) is surjective and thus is an isomorphism. Hence, \(\psi\) is also an isomorphism. ◻
We recall the following definition of \(\mathcal{P}\)-presentations for objects in \(\mathcal{H}_{\mathcal{P}}^d\) from [7].
Definition 5. For any object \(M \in \mathcal{H}_{\mathcal{P}}^d\), a \(\mathcal{P}\)-presentation of \(M\) is an object \(S\in\mathcal{P}^{[0,d]}\) satisfying \(\tau_{\geq -d+1}S \cong M\). A \(\mathcal{P}\)-presentation of a subcategory \(\mathcal{M} \subseteq \mathcal{H}_{\mathcal{P}}^d\) is a subcategory \(\mathcal{S}\subseteq\mathcal{P}^{[0,d]}\) satisfying \(\mathrm{add}(\tau_{\geq -d+1}\mathcal{S}) = \mathcal{M}\), where \(\tau_{\geq -d+1}\mathcal{S}:=\{\tau_{\geq -d+1}S \mid S\in\mathcal{S}\}\).
Lemma 2. [7]If \(\mathcal{P}\) is contravariantly finite in \(\mathcal{D}\), every object \(M\in\mathcal{H}_{\mathcal{P}}^{d}\) admits a \(\mathcal{P}\)-presentation.
Now, we can show that the truncation functor induces an equivalence of additive categories. This is a generalization of [3] and [11].
****Proposition** 6**. If \(\mathcal{P}\) is contravariantly finite in \(\mathcal{D}\), then the functor \[\tau_{\geq -d+1}:\mathcal{P}^{[0,d]}\stackrel{}{\longrightarrow}\mathcal{H}_{\mathcal{P}}^d\] induces an equivalence of additive categories \[\label{tauhz} \tau_{\geq -d+1}:\;\frac{\mathcal{P}^{[0,d]}}{\mathcal{P}[d]} \stackrel{\sim}{\longrightarrow} \mathcal{H}_{\mathcal{P}}^d.\qquad{(1)}\]
Proof. For any \(R,\,Q\in\mathcal{P}^{[0,d]}\), consider the \(t\)-decompositions of \(R\) and \(Q\): \[\begin{align} &\tau_{\leq -d}R \stackrel{}{\longrightarrow} R\stackrel{u}{\longrightarrow} \tau_{\geq -d+1}R \stackrel{}{\longrightarrow} \tau_{\leq -d}R[1],\tag{3}\\ &\tau_{\leq -d}Q \stackrel{}{\longrightarrow} Q\stackrel{v}{\longrightarrow} \tau_{\geq -d+1}Q \stackrel{}{\longrightarrow} \tau_{\leq -d}Q[1]\tag{4}. \end{align}\]
Applying \(\operatorname{Hom}(R,-)\) to the triangle 4 , we obtain an exact sequence \[\operatorname{Hom}(R,Q) \stackrel{v_\ast}{\longrightarrow} \operatorname{Hom}(R,\tau_{\geq -d+1}Q) \stackrel{}{\longrightarrow} \operatorname{Hom}(R,\tau_{\leq -d}Q[1]).\] Since \((\mathcal{P}^{[0,d]})^{\perp_{0}}\subseteq{\mathcal{P}[d]^{\perp_{\geq 0}}=\mathcal{P}^{\perp_{\geq -d}}=}\mathcal{D}^{\leq -d-1}\), we have \(\operatorname{Hom}(R,\tau_{\leq -d}Q[1]) = 0\). Thus, the map \(v_\ast\) is surjective. Note that we have the isomorphism \[\operatorname{Hom}(\tau_{\geq -d+1}R,\tau_{\geq -d+1}Q) \stackrel{u^\ast}{\longrightarrow} \operatorname{Hom}(R,\tau_{\geq -d+1}Q)\] by the adjunction \(\tau_{\geq-d+1}\dashv \sigma_{\geq-d+1}\). It is easy to see that \((u^\ast)^{-1}(v_\ast(f))=\tau_{\geq -d+1}(f)\) for any \(f\in\operatorname{Hom}(R,Q)\). Hence, we conclude that \(\tau_{\geq -d+1}\) is full.
Since \(\mathcal{P}[d] \subseteq \mathcal{P}^{\perp_{>-d}} = \mathcal{D}^{\leq -d}\), we have \(\tau_{\geq -d+1}(f)=0\) for any \(f\in\operatorname{Hom}(R,Q)\) which factors through \(\mathcal{P}{[d]}\). Conversely, let \(f\in\rm Hom(R,Q)\) such that \(\tau_{\geq -d+1}(f) = 0\). By the definition of \(\mathcal{P}^{[0,d]}\), there is a triangle \[\label{fjzhl} R^{[0,d-1]} \stackrel{\iota}{\longrightarrow} R \stackrel{\pi}{\longrightarrow} R^{d}[d]\stackrel{}{\longrightarrow} R^{[0,d-1]}[1]\tag{5}\] such that \(R^d \in \mathcal{P}\) and \(R^{[0,d-1]} \in \mathcal{P}^{[0,d-1]}\).
Applying the functor \(\operatorname{Hom}(-,Q)\) to the triangle 5 , we get an exact sequence \[\operatorname{Hom}(R^{d}[d],Q)\stackrel{\pi^\ast}{\longrightarrow} \operatorname{Hom}(R,Q)\stackrel{\iota^\ast}{\longrightarrow}\operatorname{Hom}(R^{[0,d-1]},Q).\]
Considering the \(t\)-decompositions of \(R^{[0,d-1]}\) and \(Q\) with respect to \((\mathcal{D}^{\leq-d},\mathcal{D}^{\geq-d+1})\), we obtain a commutative diagram of triangles \[\xymatrix{ \tau_{\leq-d}R^{[0,d-1]} \ar[r]\ar[d] & R^{[0,d-1]} \ar[d]^{f\circ \iota}\ar[r] & \tau_{\geq-d+1}R^{[0,d-1]}\ar[d]^{\tau_{\geq -d+1}(f\circ \iota)}\ar[r]&\tau_{\leq-d}R^{[0,d-1]}[1]\ar[d] \\ \tau_{\leq-d}Q\ar[r]&Q \ar[r]^-{p}&\tau_{\geq -d+1}Q\ar[r]&\tau_{\leq-d}Q[1]. }\] Applying \(\operatorname{Hom}(R^{[0,d-1]}, -)\) to the lower triangle, we have an exact sequence \[\operatorname{Hom}(R^{[0,d-1]}, \tau_{\le -d} Q)\stackrel{}{\longrightarrow} \operatorname{Hom}(R^{[0,d-1]}, Q) \stackrel{p_\ast}{\longrightarrow} \operatorname{Hom}(R^{[0,d-1]}, \tau_{\ge -d+1} Q).\] Since \(\mathcal{D}^{\le -d}\supseteq (\mathcal{P}^{[0,d-1]})^{ \perp_{0}},\) we have \(\operatorname{Hom}(R^{[0,d-1]}, \tau_{\le -d} Q)=0\). Thus, \(p_\ast\) is injective. Noting that \(\tau_{\geq -d+1}(f\circ \iota)=\tau_{\geq -d+1}(f)\circ \tau_{\geq -d+1}(\iota)=0,\) we get \(p_\ast(f\circ \iota)=0\). It follows that \(f\circ\iota=0\), and then \(f\) factors through \(\pi\). Thus, \(f\) factors through \(\mathcal{P}[d]\). Hence, the functor \(\tau_{\geq -d+1}:\mathcal{P}^{[0,d]}\rightarrow\mathcal{H}_{\mathcal{P}}^d\) induces the fully faithful functor ?? , which is dense by Lemma 2. Therefore, we complete the proof. ◻
As an immediate consequence of the above proposition, we have the following corollary, which is a generalization of [3].
Corollary 1. Assume that \(\mathcal{P}\) is contravariantly finite in \(\mathcal{D}\), and let \(\mathcal{V}\) be a subcategory of \(\mathcal{P}^{[0,d]}\). If \(\mathcal{V}\) is covariantly finite in \(\mathcal{P}^{[0,d]}\), then \(\tau_{\geq -d+1}\mathcal{V}\) is covariantly finite in \(\mathcal{H}_{\mathcal{P}}^d\). In addition, if \(\mathcal{P}{[d]} \subseteq \mathcal{V}\), then the converse holds.
Proof. The first statement follows directly from the fact that the functor \(\tau_{\geq -d+1}\) is full and dense. Now suppose \(\tau_{\geq -d+1}\mathcal{V}\) is covariantly finite in \(\mathcal{H}_{\mathcal{P}}^d\). For any \(R \in \mathcal{P}^{[0,d]}\), let \[\tau_{\geq -d+1}f: \tau_{\geq -d+1}R \stackrel{}{\longrightarrow} \tau_{\geq -d+1}V\] be a left \(\tau_{\geq -d+1}\mathcal{V}\)-approximation with \(V\in\mathcal{V}\).
By the structure of \(\mathcal{P}^{[0,d]}\), there is a triangle \[R^{[0,d-1]} \stackrel{\iota}{\longrightarrow} R \stackrel{\pi}{\longrightarrow} R^d[d]\stackrel{}{\longrightarrow} R^{[0,d-1]}[1]\] such that \(R^d \in \mathcal{P}\) and \(R^{[0,d-1]} \in \mathcal{P}^{[0,d-1]}\). Since \(\mathcal{P}{[d]} \subseteq \mathcal{V}\), we have \(R^d[d]\in\mathcal{V}\).
For any \(f': R \to V'\) with \(V' \in \mathcal{V}\), since \(\tau_{\geq -d+1}f\) is a left \(\tau_{\geq -d+1}\mathcal{V}\)-approximation, there exists \(\alpha: V \to V'\) such that \(\tau_{\geq -d+1}(\alpha) \circ \tau_{\geq -d+1}(f) = \tau_{\geq -d+1}(f').\) By Proposition 6, there exist morphisms \(g: R\to Q^d[d]\) and \(h:Q^d[d]\to V'\) with \(Q^d[d] \in \mathcal{P}[d]\) such that \(f'-\alpha f=h\circ g\). Since \(\operatorname{Hom}(R^{[0,d-1]}, Q^d[d]) = 0\) and then \(g\circ \iota=0\), there exists \(k:R^d[d]\rightarrow Q^d[d]\) such that \(g=k\circ\pi\), i.e. we have the following commutative diagram \[\xymatrix{ &R \ar[rr]^-{f' - \alpha f} \ar[dr]_-{g} \ar[dl]_-{\pi} & & V' \\ R^d[d] \ar[rr]_-{k} && Q^d[d]. \ar[ur]_-{h} }\] Thus, we get the following commutative diagram \[\xymatrix{ R \ar[rr]^-{\left(\begin{smallmatrix} f \\ \pi \end{smallmatrix}\right)} \ar[dr]_-{f'} & & V \oplus R^d[d] \ar[dl]^-{(\alpha \;hk)} \\ & V' }\] Hence, \({\left(\begin{smallmatrix} f \\ \pi \end{smallmatrix}\right)}: R \rightarrow V \oplus R^d[d]\) is a left \(\mathcal{V}\)-approximation of \(R\). Therefore, \(\mathcal{V}\) is covariantly finite in \(\mathcal{P}^{[0,d]}\). ◻
Let \((d+1)\)-\({\rm conf. silt}\,\mathcal{P}\) be the poset consisting of the silting subcategories of \(\mathcal{K}(\mathcal{P})\) which are contained in \(\mathcal{P}^{[0,d]}\) and contravariantly finite in \(\mathcal{D}\). Let \({\rm f.}s\operatorname{-tors}\mathcal{H}_{\mathcal{P}}^{d}\) denote the poset of functorially finite \(s\)-torsion classes in \(\mathcal{H}_{\mathcal{P}}^{d}\).
****Proposition** 7**. If \(\mathcal{P}\) is contravariantly finite in \(\mathcal{D}\), there is a monomorphism of posets \[\begin{align} \phi: (d+1)\operatorname{-conf.silt}\mathcal{P} \longrightarrow {\rm f.}s\operatorname{-tors}\mathcal{H}_{\mathcal{P}}^{d},~ \mathcal{S} \longmapsto \mathcal{S}^{\perp_{>0}} \cap \mathcal{H}_{\mathcal{P}}^d. \end{align}\]
Proof. According to [7] and its proof, we have \[\phi: (d+1)\operatorname{-conf.silt}\mathcal{P} \longrightarrow s\operatorname{-tors}\mathcal{H}_{\mathcal{P}}^{d}\] is a monomorphism of posets. It suffices to show that \(\phi (\mathcal{S})\) is covariantly finite in \(\mathcal{H}_\mathcal{P}^d\).
By Proposition 5, \(\psi (\mathcal{S})= \mathcal{S}^{\perp_{>0}} \cap \mathcal{P}^{[0,d]}\) is a hereditary cotorsion class in \(\mathcal{P}^{[0,d]}\), and thus covariantly finite in \(\mathcal{P}^{[0,d]}\).
For any \(i>0\), \(S\in\mathcal{S}\) and \(R \in \psi(\mathcal{S})\), applying \(\operatorname{Hom}(S, -)\) to the triangle \[\tau_{\le-d}R \stackrel{}{\longrightarrow} R\stackrel{}{\longrightarrow} \tau_{\ge-d+1}R \stackrel{}{\longrightarrow} \tau_{\le-d}R[1],\] we obtain the following exact sequence \[0=\operatorname{Hom}(S,\tau_{\leq -d}R[i])\to \operatorname{Hom}(S, R[i]) \to\operatorname{Hom}(S, \tau_{\ge-d+1}R[i]) \to\operatorname{Hom}(S,\tau_{\leq -d}R[i+1])=0,\] where the first and last terms vanish, since \(\mathcal{D}^{\le-d} = \mathcal{P}^{\perp_{>0}}[d] \subseteq \mathcal{S}^{\perp_{>0}}\). Thus, we have \[\operatorname{Hom}(S, \tau_{\ge-d+1}R[i]) \cong \operatorname{Hom}(S, R[i])=0.\] It follows that \(\tau_{\ge-d+1}\psi (\mathcal{S})\subseteq \phi (\mathcal{S})\). Using the same arguments and Lemma 2, we obtain \(\phi (\mathcal{S})\subseteq \tau_{\ge-d+1}\psi (\mathcal{S})\). Hence, \(\tau_{\ge-d+1}\psi (\mathcal{S}) = \phi (\mathcal{S})\). By Corollary 1, \(\phi (\mathcal{S})\) is covariantly finite in \(\mathcal{H}_{\mathcal{P}}^d\). Therefore, we complete the proof. ◻
Let us recall the Wakamatsu lemma in extriangulated categories, which will be used in the proof of the main theorem.
Lemma 3. [12]Let \(\mathcal{B}\) be an extension closed subcategory of a triangulated category \(\mathcal{C}\). Let \[X\stackrel{b}{\longrightarrow} B \stackrel{}{\longrightarrow} C \stackrel{}{\longrightarrow} X[1]\] be a triangle such that the first three terms lie in \(\mathcal{B}\) and \(b\) is a minimal left \(\mathcal{B}\)-approximation of \(X\), then \(C \in {^{\perp_1}}\mathcal{B}\).
For any \(f\in\operatorname{Hom}(X,Y)\), denote by \(C(f)\) the cone of \(f\). The following technical lemma is a generalization of [3].
Lemma 4. For any morphism \(f: R \to Q\) with \(R,\,Q\in \mathcal{P}^{[0,d]}\), we have \[\tau_{\ge-d+1}C(f) \cong \tau_{\ge-d+1}C(\tau_{\ge-d+1}f).\]
Proof. By the \(3\times3\) lemma for triangulated categories, we have the following diagram in which all rows and columns are triangles \[\xymatrix{ \tau_{\le-d}R \ar[r] \ar[d]_-{\tau_{\le-d}f} & R \ar[r] \ar[d]^f & \tau_{\ge-d+1}R \ar[d]^{\tau_{\ge-d+1}f}\ar[r]& \tau_{\le-d}R[1]\ar[d]\\ \tau_{\le-d}Q \ar[r] \ar[d] & Q \ar[r] \ar[d] & \tau_{\ge-d+1}Q \ar[d]\ar[r]&\tau_{\le-d}Q[1]\ar[d] \\ C(\tau_{\le-d}f)\ar[d] \ar[r] & C(f) \ar[d]\ar[r] & C(\tau_{\ge-d+1}f)\ar[r]\ar[d]&C(\tau_{\le-d}f)[1]\ar[d]\\ \tau_{\le-d}R[1] \ar[r] & R[1] \ar[r] & \tau_{\ge-d+1}R[1] \ar[r]& \tau_{\le-d}R[2]. }\] By the first column, we get \(C(\tau_{\le-d}f)\in\mathcal{D}^{\leq-d}\). Applying the octahedral axiom to the triangle on the third row of the above diagram and the triangle \[\tau_{\le-d}C(\tau_{\ge-d+1}f)\to C(\tau_{\ge-d+1}f)\to \tau_{\ge-d+1}C(\tau_{\ge-d+1}f)\to\tau_{\le-d}C(\tau_{\ge-d+1}f)[1],\] we obtain a commutative diagram of triangles \[\xymatrix{ C(f) \ar[r] \ar@{=}[d] & C(\tau_{\ge-d+1}f) \ar[r] \ar[d] & C(\tau_{\le-d}f)[1] \ar[d]\ar[r]&C(f)[1]\ar@{=}[d] \\ C(f) \ar[r] & \tau_{\ge-d+1}C(\tau_{\ge-d+1}f) \ar[r] \ar[d] & W[1] \ar[d]\ar[r]&C(f)[1]\ar[d]\\ & \tau_{\le-d}C(\tau_{\ge-d+1}f)[1]\ar[d] \ar@{=}[r] & \tau_{\le-d}C(\tau_{\ge-d+1}f)[1]\ar[d]\ar[r]&C(\tau_{\ge-d+1}f)[1]\\ &C(\tau_{\ge-d+1}f)[1]\ar[r]&C(\tau_{\leq-d}f)[2].& }\] By the third column of the above diagram, we have \(W[1]\in\mathcal{D}^{\leq-d-1}\). Thus, we get \[W \stackrel{}{\longrightarrow} C(f) \stackrel{}{\longrightarrow} \tau_{\ge-d+1}C(\tau_{\ge-d+1}f)\stackrel{}{\longrightarrow} W[1]\] is the \(t\)-decomposition of \(C(f)\) with respect to \((\mathcal{D}^{\leq -d},\mathcal{D}^{\geq -d+1})\). By the uniqueness of \(t\)-decomposition up to isomorphism, we finish the proof. ◻
Now, we are in a position to give the main theorem as follows.
Theorem 8. Let \(\mathcal{D}\) be a Krull-Schmidt triangulated category. Assume that all \((d+1)\)-term silting subcategories with respect to \(\mathcal{P}\) are contravariantly finite in \(\mathcal{D}\). Then there are isomorphisms of posets such that the following diagram commutes: \[\xymatrix{ (d+1)\text{-}\operatorname{silt}\mathcal{P} \ar[rr]^-{\psi}_-{\cong }\ar[dr]_-{\phi }^{\cong } && \operatorname{h.cotors}\,\mathcal{P}^{[0,d]} \ar[dl]^-{\tau_{\geq -d+1}}_{\cong } \\ & \operatorname{f.}s\operatorname{-tors}\mathcal{H}_{\mathcal{P}}^d & }\]
Proof. Note that \((d+1)\text{-}\operatorname{conf. silt}\mathcal{P}=(d+1)\text{-}\operatorname{silt}\mathcal{P}\), since all \((d+1)\)-term silting subcategories with respect to \(\mathcal{P}\) are contravariantly finite in \(\mathcal{D}\). By Propositions 5 and 7, we have the following commutative diagram \[\xymatrix{ (d+1)\text{-}\operatorname{silt}\mathcal{P} \ar[rr]^-{\psi}_{\cong }\ar@{>->}[dr]_-{\phi } && \operatorname{h.cotors}\mathcal{P}^{[0,d]} \ar[dl]^-{\tau_{\geq -d+1}} \\ & \operatorname{f.}s\operatorname{-tors}\mathcal{H}_{\mathcal{P}}^d & }\] It suffices to show that \(\tau_{\geq -d+1}:\operatorname{h.cotors}\mathcal{P}^{[0,d]}\rightarrow \operatorname{f.}s\operatorname{-tors}\mathcal{H}_{\mathcal{P}}^d\) is an epimorphism, which implies that \(\phi\) and \(\tau_{\geq -d+1}\) are isomorphisms.
For any \(\mathcal{T} \in \operatorname{f.}s\operatorname{-tors}\mathcal{H}_{\mathcal{P}}^d\), define \[\varphi(\mathcal{T}) := \left\{ V \in \mathcal{P}^{[0,d]} \mid \tau_{\geq -d+1} V \in \mathcal{T} \right\}.\] Since \(\tau_{\geq -d+1} \colon \mathcal{P}^{[0,d]} \to \mathcal{H}_{\mathcal{P}}^d\) is dense, we have \(\tau_{\geq -d+1}(\varphi(\mathcal{T})) = \mathcal{T}\). In what follows, we will prove \(\varphi(\mathcal{T}) \in \operatorname{h.cotors}\mathcal{P}^{[0,d]}\).
Step 1. \(\varphi(\mathcal{T})\) is closed under extensions.
Let \(V_1 \stackrel{f}{\longrightarrow} X \stackrel{}{\longrightarrow} V_2 \stackrel{}{\longrightarrow} V_1[1]\) be a triangle in \(\mathcal{D}\) with \(V_1,\,V_2 \in \varphi(\mathcal{T})\). By Lemma 4, we have \(\tau_{\geq -d+1} C(\tau_{\geq -d+1} f) \cong \tau_{\geq -d+1} V_2 \in \mathcal{T}.\) Let \(\mathcal{F}\) be the torsion-free class corresponding to \(\mathcal{T}\). For any \(F\in \mathcal{F}\), applying \(\operatorname{Hom}(-,F)\) to the triangle: \[\tau_{\leq -d} C(\tau_{\geq -d+1} f) \to C(\tau_{\geq -d+1} f) \to \tau_{\geq -d+1} C(\tau_{\geq -d+1} f)\to\tau_{\leq -d} C(\tau_{\geq -d+1} f)[1],\] we get \(\operatorname{Hom}(C(\tau_{\geq -d+1} f), F) = 0\). Applying \(\operatorname{Hom}(-,F)\) to the triangle \[\tau_{\geq -d+1} V_1 \xrightarrow{\tau_{\geq -d+1} f} \tau_{\geq -d+1} X \stackrel{}{\longrightarrow} C(\tau_{\geq -d+1} f)\stackrel{}{\longrightarrow}\tau_{\geq -d+1} V_1[1],\] we get \(\tau_{\geq -d+1} X \in {}^\perp\mathcal{F} \cap \mathcal{H}_{\mathcal{P}}^d = \mathcal{T}\), i.e. \(X \in \varphi(\mathcal{T})\).
Step 2. \(\varphi(\mathcal{T})\) is a cotorsion class in \(\mathcal{P}^{[0,d]}\).
Since \(\mathcal{P}[d] \in \mathcal{P}^{\perp_{>-d}} = \mathcal{D}^{\leq -d}\), we have \(\tau_{\geq -d+1}(\mathcal{P}[d]) = 0\), and then \(\mathcal{P}[d] \subseteq \varphi(\mathcal{T})\). Since \(\mathcal{T}\) is covariantly finite and \(\tau_{\geq -d+1}(\varphi(\mathcal{T})) = \mathcal{T}\), by Corollary 1 we get \(\varphi(\mathcal{T})\) is covariantly finite in \(\mathcal{P}^{[0,d]}\).
For any \(R \in \mathcal{P}^{[0,d]}\), take a minimal left \(\varphi(\mathcal{T})\)-approximation \(v \colon R \to V_R\) of \(R\). Since \(R,\,V_R \in \mathcal{P}^{[0,d]}\), we have the triangles in the two rows of the following diagram \[\label{zjfk} \xymatrix{ R^{[0,d-1]} \ar[r]^-{\iota_R} & R \ar[r]^-{\pi_R} \ar[d]^{v} & R^d[d]\ar[r]\ar@{-->}[d]^-{\alpha[d]} & R^{[0,d-1]}[1]\\ V_R^{[0,d-1]} \ar[r]^-{\iota_V} & V_R \ar[r]^-{\pi_V} & V_R^d[d]\ar[r] & R^{[0,d-1]}[1] }\tag{6}\] with \(R^{[0,d-1]},\,V_R^{[0,d-1]} \in \mathcal{P}^{[0,d-1]}\) and \(R^d,\,V_R^d \in \mathcal{P}\). Noting that \(\operatorname{Hom}(R^{[0,d-1]}, V_R^d[d]) = 0\), we have \(\pi_V\circ v\circ\iota_R=0\). Thus, there exists \(\alpha \colon R^d \to V_R^d\) such that the middle square in 6 commutes.
By [5] and the \(3\times 3\) lemma, we obtain the following diagram of triangles \[\xymatrix{ R^{[0,d-1]} \ar[r] \ar[d] & R \ar[r]^-{\pi_R} \ar[d]^-{\binom{\pi_R}{v}} & R^d[d] \ar[d]^-{\binom{id}{\alpha[d]}}\ar[r] &R^{[0,d-1]}[1]\ar[d] \\ V_R^{[0,d-1]} \ar[r]\ar[d]& R^d[d] \oplus V_R \ar[r]^-{\left(\begin{smallmatrix}id&0\\0&\pi_V\end{smallmatrix}\right)}\ar[d] & R^d[d] \oplus V_R^d[d] \ar[d]^{(-\alpha[d]~id)}\ar[r] &V_R^{[0,d-1]}[1]\ar[d] \\ Q \ar[r]\ar[d] & C(\binom{\pi_R}{v}) \ar[r]\ar[d] & V_R^d[d]\ar[r]\ar[d]&Q[1]\ar[d]\\ R^{[0,d-1]}[1] \ar[r] & R[1] \ar[r]^-{\pi_R[1]} & R^d[d+1] \ar[r] &R^{[0,d-1]}[2]. }\] Since \(Q,\,V_R^d[d] \in \mathcal{P}^{[0,d]}\), we have \(C(\binom{\pi_R}{v}) \in \mathcal{P}^{[0,d]}\). Noting that \(R^d[d] \in \mathcal{P}[d] \subseteq \varphi(\mathcal{T})\), we obtain the morphism \(\binom{\pi_R}{v}\) is also a left \(\varphi(\mathcal{T})\)-approximation of \(R\).
By the Krull-Schmidt property of \(\mathcal{D}\), there exists the following isomorphism of triangles \[\xymatrix{ R \ar[r]^-{\binom{\pi_R}{v}} \ar@{=}[d] & R^d[d] \oplus V_R \ar[r] \ar[d]^\cong & C(\binom{\pi_R}{v}) \ar[d]^\cong\ar[r]&R[1]\ar@{=}[d] \\ R \ar[r]^-{\binom{0}{v}} & R^d[d] \oplus V_R \ar[r]^{\left(\begin{smallmatrix}id&0\\0&\pi_V\end{smallmatrix}\right)} &R^d[d] \oplus C(v)\ar[r]&R[1]. }\] Thus \(C(v) \in \mathcal{P}^{[0,d]}\). By Lemma 3, we conclude \(C(v) \in {^{\perp_1}\varphi(\mathcal{T})}\).
On the other hand, we have a triangle \[R^{[1,d]}[-1] \stackrel{}{\longrightarrow} R^0\stackrel{}{\longrightarrow} R \stackrel{}{\longrightarrow} R^{[1,d]}\] such that \(R^{[1,d]}\in\mathcal{P}^{[1,d]}\) and \(R^0\in\mathcal{P}\). Since \(R^{[1,d]}[-1] \in \mathcal{P}^{[0,d]}\), by the above arguments, we have a triangle \[R^{[1,d]}[-1] \stackrel{}{\longrightarrow} V \stackrel{}{\longrightarrow} U \stackrel{}{\longrightarrow} R^{[1,d]}\] such that \(V\in\varphi(\mathcal{T})\) and \(U\in{^{\perp_1}\varphi(\mathcal{T})}\cap\mathcal{P}^{[0,d]}\). Applying the octahedral axiom, we obtain the following commutative diagram of triangles \[\xymatrix{ R[-1] \ar[r] \ar@{=}[d] & R^{[1,d]}[-1] \ar[r] \ar[d] & R^0 \ar[r] \ar[d] & R \ar@{=}[d] \\ R[-1] \ar[r] & V \ar[r] \ar[d] & W \ar[r] \ar[d] & R\ar[d] \\ & U \ar@{=}[r]\ar[d] & U\ar[r]\ar[d]&R^{[1,d]}\\ &R^{[1,d]}\ar[r]&R^0[1].& }\] Since \(R^0 \in \mathcal{P} \subseteq {^{\perp_1}\mathcal{P}^{[0,d]}} \subseteq {^{\perp_1}\varphi(\mathcal{T})}\) and \(U \in {^{\perp_1}\varphi(\mathcal{T})}\cap\mathcal{P}^{[0,d]}\), we have \(W \in {^{\perp_1}\varphi(\mathcal{T})}\cap\mathcal{P}^{[0,d]}\). Hence, \(({^{\perp_1}\varphi(\mathcal{T})}\cap\mathcal{P}^{[0,d]}, \varphi(\mathcal{T}))\) is a cotorsion pair in \(\mathcal{P}^{[0,d]}\).
Step 3. The cotorsion pair \(({^{\perp_1}\varphi(\mathcal{T})}\cap\mathcal{P}^{[0,d]}, \varphi(\mathcal{T}))\) is hereditary.
For any \(V \in \varphi(\mathcal{T})\), consider the triangle \[V^{[0,d-1]}\stackrel{}{\longrightarrow} V \stackrel{\pi_V}{\longrightarrow} V_0^d[d] \stackrel{}{\longrightarrow} V^{[0,d-1]}[1]\] with \(V^{[0,d-1]}\in\mathcal{P}^{[0,d-1]}\) and \(V_0^d\in\mathcal{P}\). Since \(V_0^d[d]\in \mathcal{P}[d]\subseteq\mathcal{D}^{\leq -d}\), we have \[C(\tau_{\geq -d+1}\pi_V)\cong (\tau_{\geq -d+1} V)[1]\in \mathcal{T}[1].\] Noting that \((\mathcal{T},\mathcal{F})\) is an \(s\)-torsion pair in \(\mathcal{H}_{\mathcal{P}}^d\), we have \(\operatorname{Hom}(C(\tau_{\geq -d+1}\pi_V), \mathcal{F}) = 0\).
For any \(F\in\mathcal{F}\), applying \(\operatorname{Hom}(-,F)\) to the triangle \[\tau_{\leq -d} C(\tau_{\geq -d+1}\pi_V) \to C(\tau_{\geq -d+1}\pi_V) \to \tau_{\geq -d+1} C(\tau_{\geq -d+1}\pi_V)\to\tau_{\leq -d} C(\tau_{\geq -d+1}\pi_V)[1],\] we obtain an exact sequence \[\operatorname{Hom}(\tau_{\leq -d} C(\tau_{\geq -d+1}\pi_V)[1], F) \to \operatorname{Hom}(\tau_{\geq -d+1} C(\tau_{\geq -d+1}\pi_V), F) \to \operatorname{Hom}(C(\tau_{\geq -d+1}\pi_V), F).\] Since \(\operatorname{Hom}(\tau_{\leq -d} C(\tau_{\geq -d+1}\pi_V)[1], F)=0\) and \(\operatorname{Hom}(C(\tau_{\geq -d+1}\pi_V), F) = 0\), we get \[\tau_{\geq -d+1} V^{[0,d-1]}[1] \cong \tau_{\geq -d+1} C(\tau_{\geq -d+1}\pi_V) \in \mathcal{T}.\] Thus, we obtain a triangle \[V^{[0,d-1]}[1] \stackrel{}{\longrightarrow} V[1] \stackrel{}{\longrightarrow} V_0^d[d+1] \stackrel{}{\longrightarrow} V^{[0,d-1]}[2]\] with \(V^{[0,d-1]}[1] \in \varphi(\mathcal{T})\) and \(V_0^d[d+1] \in \mathcal{P}[d+1]\). Furthermore, for \(V^{[0,d-1]}[1]\) we have the triangle \[V^{[1,d-1]}\stackrel{}{\longrightarrow} V^{[0,d-1]}[1] \stackrel{}{\longrightarrow} V_1^d[d] \stackrel{}{\longrightarrow} V^{[1,d-1]}[1]\] with \(V^{[1,d-1]}\in\mathcal{P}^{[1,d-1]}\) and \(V_1^d\in\mathcal{P}\). By the same argument, we get \(V^{[1,d-1]}[1] \in \varphi(\mathcal{T})\). Applying the octahedral axiom yields the following commutative diagram of triangles \[\xymatrix{ V^{[1,d-1]} \ar@{=}[d] \ar[r] & V^{[0,d-1]}[1] \ar[r] \ar[d] & V_1^d[d] \ar[d]\ar[r]&V^{[1,d-1]}[1]\ar@{=}[d] \\ V^{[1,d-1]} \ar[r] & V[1] \ar[r] \ar[d] & V^{[d,d+1]} \ar[d]\ar[r]&V^{[1,d-1]}[1] \ar[d]\\ & V_0^d[d+1] \ar@{=}[r]\ar[d] & V_0^d[d+1]\ar[r]\ar[d]&V^{[0,d-1]}[2]\\ &V^{[0,d-1]}[2]\ar[r]&V_1^d[d+1].& }\] By the triangle in the second row of the above diagram, we have the following triangle \[V^{[1,d-1]}[1] \stackrel{}{\longrightarrow} V[2] \stackrel{}{\longrightarrow} V^{[d,d+1]}[1] \stackrel{}{\longrightarrow} V^{[1,d-1]}[2]\] with \(V^{[1,d-1]}[1] \in \varphi(\mathcal{T})\) and \(V^{[d,d+1]}[1] \in \mathcal{P}^{[d+1,d+2]}\).
By induction, for each integer \(0 \leq i \leq d-1\), we have the following triangle \[V^{[i,d-1]}[1] \stackrel{}{\longrightarrow} V[i+1] \stackrel{}{\longrightarrow} V^{[d,d+i]}[1] \stackrel{}{\longrightarrow} V^{[i,d-1]}[2]\] with \(V^{[i,d-1]}[1] \in \varphi(\mathcal{T})\) and \(V^{[d,d+i]}[1] \in \mathcal{P}^{[d+1,d+i+1]}\).
For any \(U \in {^{\perp_1}\varphi(\mathcal{T})}\cap\mathcal{P}^{[0,d]}\), applying \(\operatorname{Hom}(U, -)\) to the above triangles yields the exact sequences \[0=\operatorname{Hom}(U, V^{[i,d-1]}[2]) \stackrel{}{\longrightarrow} \operatorname{Hom}(U, V[i+2]) \stackrel{}{\longrightarrow} \operatorname{Hom}(U, V^{[d,d+i]}[2]) = 0.\] Note that \(V[j+2]\in \mathcal{P}^{[d+2,+\infty)}\subseteq (\mathcal{P}^{[0,d]})^{\perp_0}\) for any \(j \geq d\). So, \(\operatorname{Hom}(U, V[\geq 2])=0\). Hence, \(\operatorname{Hom}\big({^{\perp_1}\varphi(\mathcal{T})}\cap\mathcal{P}^{[0,d]},\;\varphi(\mathcal{T})[\geq 2]\big)=0,\) which means the cotorsion pair is hereditary. ◻
-0.5Applications to DG algebras In this section, let \(A\) be a non-positive dg algebra over a field \(K\) whose total cohomology \(H^*(A)\) is finite dimensional. Denote by \(D(A)\) the derived category of (right) dg \(A\)-modules. Let \(D_{\rm fd}(A)\) be the full subcategory of \(D(A)\) consisting of dg \(A\)-modules \(M\) with finite-dimensional total cohomology \(H^\ast(M)\). Then \(A_A \in D_{\rm fd}(A)\) and \(D_{\rm fd}(A)\) admits a standard \(t\)-structure \((D_{\rm fd}(A)^{\leq 0}, D_{\rm fd}(A)^{\geq 1})\), where \[D_{\rm fd}(A)^{\leq 0} := \{ M \in D_{\rm fd}(A) \mid H^i(M) = 0 \text{ for all } i>0 \} = A^{\perp_{>0}}\] and \[D_{\rm fd}(A)^{\geq 1} := \{ M \in D_{\rm fd}(A) \mid H^i(M) = 0 \text{ for all } i \leq 0 \} = A^{\perp_{\leq 0}}.\] The heart of this \(t\)-structure is equivalent to the category \(\mathrm{mod}(H^0(A))\) of finite dimensional modules over \(H^0(A)\).
By [8], \({D}_{\rm fd}(A)\) is Hom-finite. Note that \(D_{\rm fd}(A)\) is closed under direct summands in \({D}(A)\) and then \({D}_{\rm fd}(A)\) is idempotent complete. Thus, \({D}_{\rm fd}(A)\) is a \(K\)-linear Hom-finite Krull-Schmidt triangulated category. Therefore, the subcategories \(\mathcal{H}_A^d:=D_{\rm fd}(A)^{\leq 0} \cap {D}_{\rm fd}(A)^{\geq -d+1}\) and \(\mathrm{per}(A):={\boldsymbol{t}hick}_{D_{\rm fd}(A)}(A)\) of \({D}_{\rm fd}(A)\) are also Krull-Schmidt.
Since \(A\) is a silting object of \(\mathrm{per}(A)\), i.e. the additive closure \(\mathrm{add}(A)\) of \(A\) in \(\mathrm{per}(A)\) is a silting subcategory of \(\mathrm{per}(A)\), by [13], any silting subcategory of \(\mathrm{per}(A)\) is of the form \(\mathrm{add}(T)\) for some object \(T\in\mathrm{per}(A)\), and thus is contravariantly finite in \({D}_{\rm fd}(A)\). Since \(H^*(A)\) is finite dimensional, there exists a positive integer \(d\) such that \(A \in \mathcal{H}_A^d\).
Set \((d+1)\)-\(\mathrm{silt}\,A := (d+1)\)-\(\mathrm{silt}\,\mathrm{add}(A)\). As an application of Theorem 8, we have the following.
Corollary 2. There are isomorphisms of posets such that the following diagram commutes \[\xymatrix{ (d+1)\text{-}\mathrm{silt}\,A \ar[rr]^{\cong} \ar[rd]_{\cong} & & \operatorname{h.cotors}\,\mathrm{add}(A)^{[0,d]} \ar[ld]^{\cong} \\ & \operatorname{f.}s\operatorname{-tors}\mathcal{H}_A^d & }\]
For convenience, let us briefly recall the construction of an \(\mathrm{add}(A)\)-presentation for any \(M\in\mathcal{H}_A^d\). One may refer to [7] and [11] for more details.
For any \(M \in \mathcal{H}_A^d\) and \(0\leq i\leq d\), we have the following triangles in \({D}_{\rm fd}(A)\): \[\Omega^{i+1}(M) \stackrel{\iota^i}{\longrightarrow} P^i \stackrel{\pi^i}{\longrightarrow} \Omega^i(M) \stackrel{h^i}{\longrightarrow}\Omega^{i+1}(M)[1],\] where \(\Omega^0(M)=M\) and \(\pi^i\) is the minimal right \(\mathrm{add}(A)\)-approximation of \(\Omega^i(M)\).
Let \(h:= h^d[d] \circ \cdots \circ h^1[1] \circ h^0:M \to \Omega^{d+1}(M)[d+1]\). According to the proof of [11], applying the octahedral axiom inductively, we finally obtain the following commutative diagram of triangles \[\label{ppst} \xymatrix{& P^{[0,d-1]} \ar@{=}[r] \ar[d] & P^{[0,d-1]} \ar[d] \\ \Omega^{d+1}(M)[d] \ar[r] \ar@{=}[d] & p(M) \ar[r] \ar[d] & M \ar[r]^-h\ar[d] & \Omega^{d+1}(M)[d+1] \ar@{=}[d] \\ \Omega^{d+1}(M)[d] \ar[r]^-{\iota^d[d]} & P^d[d] \ar[r]^-{\pi^d[d]}\ar[d] & \Omega^d(M)[d] \ar[r]^-{h^d[d]}\ar[d] & \Omega^{d+1}(M)[d+1]\ar[d]\\ &P^{[0,d-1]}[1] \ar@{=}[r] & P^{[0,d-1]}[1]\ar[r]&p(M)[1]}\tag{7}\] where \(P^{[0,d-1]}\in \mathrm{add}(A)^{[0,d-1]}\) and \(p(M)\) is an \(\mathrm{add}(A)\)-presentation of \(M\).
****Remark** 9**. We claim that the \(p(M)\) in 7 does not have \(P[d]\) as direct summands for any nonzero \(P \in \mathrm{add}(A)\). Otherwise, since \(\mathrm{Hom}(P^{[0,d-1]}, P[d]) = 0\) and \(\mathrm{Hom}(P[d], M) = 0\), the morphisms \(p(M) \to P^d[d]\) and \(\Omega^{d+1}(M)[d] \to p(M)\) in 7 have \(P[d] \xrightarrow{id} P[d]\) as direct summands. Then \(\iota^d[d]\) has \(P[d] \xrightarrow{id} P[d]\) as direct summands. This contradicts the right minimality of \(\pi^d[d]\). Furthermore, by Proposition 6, \(p(M)\) is unique up to isomorphism. In particular, \(p(M)\) and \(M\) have the same number of indecomposable direct summands, which is denoted by \(|p(M)| = |M|\). In what follows, for each \(M \in \mathcal{H}_A^d\), we fix the notation \(p(M)\) for the \(\mathrm{add}(A)\)-presentation of \(M\) given in 7 .
An object \(P\) of \(\mathcal{H}_A^d\) is called projective if \(\mathrm{Hom}(P, M[1]) = 0\) for any \(M \in \mathcal{H}_A^d\). Dually, an object \(I\) of \(\mathcal{H}_A^d\) is called injective if \(\mathrm{Hom}(M, I[1]) = 0\) for any \(M \in \mathcal{H}_A^d\). Denote by \(\mathcal{P}(\mathcal{H}_A^d)\) and \(\mathcal{I}(\mathcal{H}_A^d)\) the full subcategories of projective and injective objects in \(\mathcal{H}_A^d\), respectively. Let \(\underline{\mathcal{H}_A^d}:=\mathcal{H}_A^d/\mathcal{P}(\mathcal{H}_A^d)\) and \(\overline{\mathcal{H}_A^d}:=\mathcal{H}_A^d/\mathcal{I}(\mathcal{H}_A^d)\). The category equivalence \(\tau:\underline{\mathcal{H}_A^d}\to\overline{\mathcal{H}_A^d}\) defined in [11] maps \(M\) to \(\tau_{\leq 0}(\nu (p(M))[-1])\), where \(\nu:D_{\rm fd}(A)\to D_{\rm fd}(A)\) denotes the derived Nakayama functor. Analogous to [4], we have the following.
Lemma 5. Let \(M,N \in \mathcal{H}_A^d\) and \(P \in \mathrm{add}(A)\). For any \(i\geq 1\), there are isomorphisms \[\mathrm{Hom}(p(M), p(N)[i]) \cong \mathrm{Hom}(p(M), N[i])\cong \mathbb{D}\mathrm{Hom}(N, \tau(M)[1-i])\] and \[\mathrm{Hom}(P[d], p(N)[i]) \cong \mathrm{Hom}(P, N[i-d]),\] where \(\mathbb{D}:=\mathrm{Hom}_K(-,K)\) denotes the \(K\)-linear duality.
Following [4], we generalize the \(\tau\)-tilting theory in \(\mathrm{mod}(H^0(A))\) to \(\mathcal{H}_A^d\).
Definition 6. [4] Let \(\mathcal{M}\) be a subcategory of \(\mathcal{H}^d_A\), and let \(n\) be a positive integer. Suppose there exist extriangles in \(\mathcal{H}^d_A\) \[\label{ejiao} X_{i+1}\stackrel{}{\longrightarrow}M_{i+1}\stackrel{}{\longrightarrow}X_i\stackrel{}\dashrightarrow\tag{8}\] with \(M_{i+1} \in \mathcal{M}\), \(0\leq i\leq n-1\). Then \(X_0\) is called an \(n\)-factor of \(\mathcal{M}\). We denote by \(\mathrm{Fac}_n(\mathcal{M})\) the subcategory of \(\mathcal{H}_A^d\) consisting of all \(n\)-factors of \(\mathcal{M}\).
Definition 7. [4] (a) A dg \(A\)-module \(M\in\mathcal{H}_A^d\) is called positive \(\tau\)-rigid if \(\mathrm{Hom}(M, \tau(M)[\leq 0]) = 0.\)
(b) A pair \((M,P)\) of \(M \in \mathcal{H}_A^d\) and \(P \in \mathrm{add}(A)\) is called a positive \(\tau\)-rigid pair if \(M\) is positive \(\tau\)-rigid and \(\mathrm{Hom}(P, M[\leq 0]) = 0.\)
(c) A positive \(\tau\)-rigid pair \((M,P)\) is called \(\tau\)-tilting if \[{}^{\perp_{\leq 0}}(\tau(M)) \cap P^{\perp_{\leq 0}} \cap \mathcal{H}_A^d =\mathrm{Fac}_d(\mathrm{add}(M))=:\mathrm{Fac}_d(M).\] An object in \(\mathcal{H}_A^d\) is called basic, if its indecomposable direct summands are pairwise non-isomorphic. A positive \(\tau\)-rigid pair or \(\tau\)-tilting pair \((M,P)\) is said to be basic, if \(M\) and \(P\) are basic. We denote by \(\tau\text{-}\mathrm{tiltp}\,\mathcal{H}_A^d\) the set consisting of all basic \(\tau\)-tilting pairs in \(\mathcal{H}^d_A\). For any \((M_1,P_1),(M_2,P_2)\in \tau\text{-}\mathrm{tiltp}\,\mathcal{H}_A^d\), define \((M_1,P_1)\leq(M_2,P_2)\) if \(\mathrm{Fac}_d(M_1)\subseteq \mathrm{Fac}_d(M_2)\).
****Remark** 10**. Let \((M,P)\) be a positive \(\tau\)-rigid pair of \(\mathcal{H}_A^d\). By [4], we have \(\mathrm{Fac}_d(M)\subseteq {^{\perp_{\leq 0}}(\tau(M))}.\) By the definition of \(\mathrm{Fac}_d(M)\) and applying \(\operatorname{Hom}(P,-)\) to the extriangles in 8 , we get \(\mathrm{Fac}_d(M)\subseteq P^{\perp_{[-d+1,0]}}.\) Since \(\mathrm{Fac}_d(M)\subseteq \mathcal{H}_A^d\subseteq A^{\perp_{\leq -d}},\) we have \(\mathrm{Fac}_d(M)\subseteq{}^{\perp_{\leq 0}}(\tau(M)) \cap P^{\perp_{\leq 0}} \cap \mathcal{H}_A^d.\) Thus, a positive \(\tau\)-rigid pair \((M,P)\) is \(\tau\)-tilting if and only if \({}^{\perp_{\leq 0}}(\tau(M)) \cap P^{\perp_{\leq 0}} \cap \mathcal{H}_A^d \subseteq \mathrm{Fac}_d(M).\)
Definition 8. [7] Let \(\mathcal{M}\) be a subcategory of \(\mathcal{H}_A^d\). If it admits an \(\mathrm{add}(A)\)-presentation \(\mathcal{S}\) such that \(\mathcal{S}^{\perp_{>0}}\cap \mathcal{H}_A^d=\mathrm{Fac}_d(\mathcal{M}),\) then \(\mathcal{M}\) is called an AIR tilting subcategory of \(\mathcal{H}_A^d\) with respect to \(\mathcal{S}\). We denote by \(\mathrm{AIR}\text{-}\mathrm{tilt}\,\mathcal{H}_A^d\) the collection consisting of all AIR tilting subcategories of \(\mathcal{H}^d_A\). Similar to \(\tau\text{-}\mathrm{tiltp}\,\mathcal{H}_A^d\), a partial order relation can be defined on \(\mathrm{AIR}\text{-}\mathrm{tilt}\,\mathcal{H}_A^d\).
Lemma 6. [7] Let \(\mathcal{M}\) be a subcategory of \(\mathcal{H}_A^d\). If \(\mathcal{M}\) is AIR tilting with respect to its two \(\mathrm{add}(A)\)-presentations \(\mathcal{S}\) and \(\mathcal{S}'\), then \(\mathcal{S}=\mathcal{S}'\).
****Proposition** 11**. There exist isomorphisms of posets such that the following diagram commutes \[\label{diyihang} \xymatrix{ \mathrm{AIR}\text{-}\mathrm{tilt}\,\mathcal{H}_A^d \ar[rr]^{\cong} && (d+1)\text{-}\mathrm{silt}\,A \\ &\tau\text{-}\mathrm{tiltp}\,\mathcal{H}_A^d \ar[ru]_{\cong} \ar[lu]^{\cong} & }\qquad{(2)}\]
Proof. The isomorphism in the first row of ?? follows from [7], which maps the AIR-tilting subcategories of \(\mathcal{H}_A^d\) to their \(\mathrm{add}(A)\)-presentations.
Let \((M,P)\) be a pair with \(M \in \mathcal{H}_A^d\) and \(P \in \mathrm{add}(A)\). Set \(\mathbb{P}:= p(M) \oplus P[d]\). By Proposition 6, we get \(\mathrm{add}(\mathbb{P})\) is an \(\mathrm{add}(A)\)-presentation of \(\mathrm{add}(M)\).
Note that for any \(N \in \mathcal{H}_A^d= A^{\perp_{\leq -d}}\cap A^{\perp_{>0}}=A[d]^{\perp_{\leq 0}}\cap A[d]^{\perp_{>d}}\), \[\mathrm{Hom}(P[d], N[i]) = 0 \text{ for any } i \geq 1\] if and only if \[\mathrm{Hom}(P[d], N[i]) = 0 \text{ for any } i \leq d\] if and only if \[\mathrm{Hom}(P, N[i]) = 0 \text{ for any } i \leq 0.\] Thus, we have \(P^{\perp_{\leq 0}}\cap \mathcal{H}_A^d = P[d]^{\perp_{>0}}\cap \mathcal{H}_A^d\). By Lemma 5, we have \({}^{\perp_{\leq 0}}(\tau(M))\cap \mathcal{H}_A^d = p(M)^{\perp_{>0}}\cap \mathcal{H}_A^d\). Hence, we get \({}^{\perp_{\leq 0}}(\tau M) \cap P^{\perp_{\leq 0}}{\cap \mathcal{H}_A^d} = \mathbb{P}^{\perp_{>0}}\cap \mathcal{H}_A^d\), and it follows that \((M,P)\) forms a positive \(\tau\)-rigid pair if and only if \(\mathrm{Hom}(\mathbb{P},M[>0])=0.\)
Moreover, if \((M,P)\) is a basic \(\tau\)-tilting pair, then \[\mathrm{Fac}_d(M) = {}^{\perp_{\leq 0}}(\tau M) \cap P^{\perp_{\leq 0}} \cap \mathcal{H}_A^d = \mathbb{P}^{\perp_{>0}} \cap \mathcal{H}_A^d.\] It follows that the subcategory \(\mathrm{add}(M)\) of \(\mathcal{H}_A^d\) is AIR tilting with respect to its \(\mathrm{add}(A)\)-presentation \(\mathrm{add}(\mathbb{P})\).
If \((M,Q)\) is also a basic \(\tau\)-tilting pair, then \((M,P\oplus Q)\) is a positive \(\tau\)-rigid pair. Since \((M,Q)\) is a \(\tau\)-tilting pair, we obtain \[{}^{\perp_{\leq 0}}(\tau(M)) \cap (P\oplus Q)^{\perp_{\leq 0}} \cap \mathcal{H}_A^d\subseteq{}^{\perp_{\leq 0}}(\tau(M)) \cap Q^{\perp_{\leq 0}} \cap \mathcal{H}_A^d\subseteq\mathrm{Fac}_d(M).\] By Remark 10, \((M,P\oplus Q)\) is a \(\tau\)-tilting pair. According to the above discussion, we obtain \(p(M)\oplus P[d]\oplus Q[d]\) is a silting object of \(\mathrm{per}(A)\). By [13], we get \[|p(M)\oplus (P\oplus Q)[d]| = |\mathbb{P}|=|p(M)\oplus P[d]|.\] Since \(p(M)\) does not have \(P'[d]\) as direct summands for any nonzero \(P' \in \mathrm{add}(A)\), we get \(|P\oplus Q|=|P|\). Hence, \(Q\) is a direct summand of \(P\). Similarly, \(P\) is also a direct summand of \(Q\). Since \(P\) and \(Q\) are both basic, we conclude \(P\cong Q\). Moreover, since \(M\) is basic, we get that the correspondence \((M,P)\mapsto \mathrm{add}(M)\) gives an injective map from \(\tau\text{-}\mathrm{tiltp}\,\mathcal{H}_A^d\) to \(\mathrm{AIR}\text{-}\mathrm{tilt}\,\mathcal{H}_A^d\), and thus an injective map from \(\tau\text{-}\mathrm{tiltp}\,\mathcal{H}_A^d\) to \((d+1)\text{-}\mathrm{silt}\,A\) by combining with the isomorphism in the first row of ?? .
Conversely, let \(\mathcal{S}\) be a silting subcategory of \(\mathrm{per}(A)\). By [13] and the Krull-Schmidt property of \(\operatorname{per}(A)\), we can take its unique basic additive generator \(\mathbb{S} = S \oplus P[d]\), where \(P \in \mathrm{add}(A)\) and \(S\) does not have \(P'[m]\) as direct summands for any nonzero \(P' \in \mathrm{add}(A)\), then \(S = p(\tau_{\geq -d+1}\mathbb{S})\). By Lemma 5, we have \[\mathrm{Hom}(P,\tau_{\geq-d+1}\mathbb{S}[i-d])\cong\mathrm{Hom}(P[d],S[i])=0\] for any \(1\leq i\leq d\), where the last equality follows from \(\mathbb{S}\) being silting. Moreover, since \(\tau_{\geq-d+1}\mathbb{S}\in A^{\perp_{\leq-d}}\), we obtain \((\tau_{\geq -d+1}\mathbb{S}, P)\) is a basic positive \(\tau\)-rigid pair.
Noting that \(P\) is a \((d+1)\)-term silting subcategory of \(\mathrm{per}(A)\), by [7], we get \[\mathrm{Fac}_d(\tau_{\geq-d+1}\mathbb{S}) = \mathbb{S}^{\perp_{>0}}\cap \mathcal{H}_A^d = {}^{\perp_{\leq 0}}(\tau (\tau_{\geq -d+1}\mathbb{S})) \cap P^{\perp_{ \leq 0}} \cap \mathcal{H}_A^d.\] Hence, \((\tau_{\geq-d+1}\mathbb{S}, P)\) is a basic \(\tau\)-tilting pair, and then \(\mathrm{add}(\tau_{\geq-d+1}\mathbb{S})\) is the AIR tilting subcategory with respect to its \(\mathrm{add}(A)\)-presentation \(\mathrm{add}(\mathbb{S})=\mathcal{S}\). Therefore, we complete the proof. ◻
-0.5*Acknowledgments This work is partially supported by the National Natural Science Foundation of China (No. 12271257) and the Natural Science Foundation of Jiangsu Province of China (No. BK20240137).